local extremum

A local max or min — a point that beats only its immediate neighbors, not necessarily the whole domain.

$f$ has a local maximum at $(a,b)$ if $f(a,b)\geq f(x,y)$ for every $(x,y)$ in some small disk around $(a,b)$ (a local minimum reverses the inequality) — a purely neighborhood-scale comparison the second derivative test can certify directly. A local extremum only has to beat nearby points; a domain can easily contain another point, arbitrarily far away, that beats it, which is exactly why a local extremum need not be an absolute one.

Examples

For $f(x,y)=x^2+y^2$ restricted to the punctured domain $(x,y)\neq(0,0)$ , points arbitrarily close to the origin still show no local extremum exists there, since the point that would attain it was removed.